Johannes Wallner

Geometric modeling with conical meshes and developable surfaces

In architectural freeform design, the relation between shape and fabrication poses new challenges and requires more sophistication from the underlying geometry. The new concept of conical meshes satisfies central requirements for this application: They are quadrilateral meshes with planar faces, and therefore particularly suitable for the design of freeform glass structures. Moreover, they possess a natural offsetting operation and provide a support structure orthogonal to the mesh. Being a discrete analogue of the network of principal curvature lines, they represent fundamental shape characteristics. We show how to optimize a quad mesh such that its faces become planar, or the mesh becomes even conical. Combining this perturbation with subdivision yields a powerful new modeling tool for all types of quad meshes with planar faces, making subdivision attractive for architecture design and providing an elegant way of modeling developable surfaces.

Discovering structural regularity in 3D geometry

We introduce a computational framework for discovering regular or repeated geometric structures in 3D shapes. We describe and classify possible regular structures and present an effective algorithm for detecting such repeated geometric patterns in point- or meshbased models. Our method assumes no prior knowledge of the geometry or spatial location of the individual elements that define the pattern. Structure discovery is made possible by a careful analysis of pairwise similarity transformations that reveals prominent lattice structures in a suitable model of transformation space. We introduce an optimization method for detecting such uniform grids specifically designed to deal with outliers and missing elements. This yields a robust algorithm that successfully discovers complex regular structures amidst clutter, noise, and missing geometry. The accuracy of the extracted generating transformations is further improved using a novel simultaneous registration method in the spatial domain. We demonstrate the effectiveness of our algorithm on a variety of examples and show applications to compression, model repair, and geometry synthesis.

Integral invariants for robust geometry processing

Differential invariants of curves and surfaces such as curvatures and their derivatives play a central role in Geometry Processing. They are, however, sensitive to noise and minor perturbations and do not exhibit the desired multi-scale behavior. Recently, the relationships between differential invariants and certain integrals over small neighborhoods have been used to define efficiently computable integral invariants which have both a geometric meaning and useful stability properties. This paper considers integral invariants defined via distance functions, and the stability analysis of integral invariants in general. Such invariants proved useful for many tasks where the computation of shape characteristics is important. A prominent and recent example is the automatic reassembling of broken objects based on correspondences between fracture surfaces.

Geometry of multi-layer freeform structures for architecture

The geometric challenges in the architectural design of freeform shapes come mainly from the physical realization of beams and nodes. We approach them via the concept of parallel meshes, and present methods of computation and optimization. We discuss planar faces, beams of controlled height, node geometry, and multilayer constructions. Beams of constant height are achieved with the new type of edge offset meshes. Mesh parallelism is also the main ingredient in a novel discrete theory of curvatures. These methods are applied to the construction of quadrilateral, pentagonal and hexagonal meshes, discrete minimal surfaces, discrete constant mean curvature surfaces, and their geometric transforms. We show how to design geometrically optimal shapes, and how to find a meaningful meshing and beam layout for existing shapes.

Approximation algorithms for developable surfaces

Abstract By its dual representation, a developable surface can be viewed as a curve of dual projective3-space. After introducing an appropriate metric in the dual space and restricting ourselves to special parametrizations of the surfaces involved, we derive linear approximation algorithms for developable NURBS surfaces, including multiscale approximations. Special attention is paid to controlling the curve of regression.

Freeform surfaces from single curved panels

Motivated by applications in architecture and manufacturing, we discuss the problem of covering a freeform surface by single curved panels. This leads to the new concept of semi-discrete surface representation, which constitutes a link between smooth and discrete surfaces. The basic entity we are working with is the developable strip model. It is the semi-discrete equivalent of a quad mesh with planar faces, or a conjugate parametrization of a smooth surface. We present a B-spline based optimization framework for efficient computing with D-strip models. In particular we study conical and circular models, which semi-discretize the network of principal curvature lines, and which enjoy elegant geometric properties. Together with geodesic models and cylindrical models they offer a rich source of solutions for surface panelization problems.

Convergence and C1 analysis of subdivision schemes on manifolds by proximity

Curve subdivision schemes on manifolds and in Lie groups are constructed from linear subdivision schemes by first representing the rules of affinely invariant linear schemes in terms of repeated affine averages, and then replacing the operation of affine average either by a geodesic average (in the Riemannian sense or in a certain Lie group sense), or by projection of the affine averages onto a surface. The analysis of these schemes is based on their proximity to the linear schemes which they are derived from. We verify that a linear scheme S and its analogous nonlinear scheme T satisfy a proximity condition. We further show that the proximity condition implies the convergence of T and continuity of its limit curves, if S has the same property, and if the distances of consecutive points of the initial control polygon are small enough. Moreover, if S satisfies a smoothness condition which is sufficient for its limit curves to be C^1, and if T is convergent, then the curves generated by T are also C^1. Similar analysis of C^2 smoothness is postponed to a forthcoming paper.

Robust Feature Classification and Editing

Sharp edges, ridges, valleys, and prongs are critical for the appearance and an accurate representation of a 3D model. In this paper, we propose a novel approach that deals with the global shape of features in a robust way. Based on a remeshing algorithm which delivers an isotropic mesh in a feature-sensitive metric, features are recognized on multiple scales via integral invariants of local neighborhoods. Morphological and smoothing operations are then used for feature region extraction and classification into basic types such as ridges, valleys, and prongs. The resulting representation of feature regions is further used for feature-specific editing operations

Design of self-supporting surfaces

Self-supporting masonry is one of the most ancient and elegant techniques for building curved shapes. Because of the very geometric nature of their failure, analyzing and modeling such strutures is more a geometry processing problem than one of classical continuum mechanics. This paper uses the thrust network method of analysis and presents an iterative nonlinear optimization algorithm for efficiently approximating freeform shapes by self-supporting ones. The rich geometry of thrust networks leads us to close connections between diverse topics in discrete differential geometry, such as a finite-element discretization of the Airy stress potential, perfect graph Laplacians, and computing admissible loads via curvatures of polyhedral surfaces. This geometric viewpoint allows us, in particular, to remesh self-supporting shapes by self-supporting quad meshes with planar faces, and leads to another application of the theory: steel/glass constructions with low moments in nodes.

On Surface Approximation Using Developable Surfaces

We introduce a method for approximating a given surface by a developable surface. It will be either aG1surface consisting of pieces of cones or cylinders of revolution or aGrNURBS developable surface. Our algorithm will also deal properly with the problems of reverse engineering and produce robust approximation of given scattered data. The presented technique can be applied in computer aided manufacturing, e.g. in shipbuilding.